A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the [alpha]BB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained optimization problem. Moreover, the multiobjective reformulation enables to identify the trade-off between constraint satisfaction and objective value which is also reflected in the quality guarantee. Numerical tests validate that we indeed can find feasible and often optimal solutions where the classical single-objective [alpha]BB method fails, i.e., it terminates without ever finding a feasible solution.